Solve the initial-value problem for y as a function of x
(x^2+49)dy/dx=1 ,y(7)=0
come on . we just did this
1/7 arctan(7/7) + C = 0
now just solve for C.
BRUH MOMENTO
To solve the initial-value problem for y as a function of x, we'll integrate both sides of the differential equation and then apply the initial condition.
Step 1: Rewrite the differential equation:
We have (x^2 + 49)dy/dx = 1. Rearranging the equation, we get dy = dx / (x^2 + 49).
Step 2: Integrate both sides:
Integrating both sides of the equation, we have ∫dy = ∫dx / (x^2 + 49).
This gives us y = ∫(1 / (x^2 + 49))dx.
Step 3: Evaluate the integral:
To integrate 1 / (x^2 + 49), we can use the substitution method. Let's substitute u = x/7. Then, du = dx/7, and dx = 7 du.
Now, our integral becomes y = ∫(1 / (49u^2 + 49))(7 du) = 7∫(1 / (u^2 + 1))du.
This integral is a standard integral and can be written as 7 * arctan(u) + C, where C is the constant of integration.
Step 4: Apply the initial condition:
y(7) = 0, which means when x = 7, y = 0.
Substituting these values into the solution equation, we have:
0 = 7 * arctan(7/7) + C.
0 = 7 * arctan(1) + C.
0 = 7 * (π/4) + C.
0 = (7π/4) + C.
C = -(7π/4).
Step 5: Final solution:
Now that we have the value of C, we can write the final solution:
y = 7 * arctan(x/7) - (7π/4).
Therefore, the solution to the initial-value problem is y = 7 * arctan(x/7) - (7π/4).
To solve the initial-value problem for y as a function of x, we'll follow these steps:
Step 1: Rewrite the given differential equation in the form dy/dx = f(x, y).
Step 2: Determine the general solution to the differential equation.
Step 3: Apply the initial condition to find the particular solution.
Step 4: Write the final solution.
Now, let's go through each step in detail:
Step 1: Rewrite the given differential equation in the form dy/dx = f(x, y).
The given differential equation is (x^2 + 49) dy/dx = 1. To rewrite it in the desired form, we divide both sides by (x^2 + 49):
dy/dx = 1 / (x^2 + 49)
Step 2: Determine the general solution to the differential equation.
To find the general solution, we need to integrate both sides of the equation with respect to x:
∫dy = ∫(1 / (x^2 + 49)) dx
The integral of 1 / (x^2 + 49) is arctan(x/7) + C, where C is the constant of integration.
So, the general solution to the differential equation is y = arctan(x/7) + C.
Step 3: Apply the initial condition to find the particular solution.
We are given the initial condition y(7) = 0. Substituting the values into the general solution, we have:
0 = arctan(7/7) + C
0 = arctan(1) + C
The arctan(1) is π/4, so we have:
0 = π/4 + C
Solving for C, we subtract π/4 from both sides:
C = -π/4
Step 4: Write the final solution.
Now that we have the value of C, we can write the particular solution. Plugging in C = -π/4 into the general solution, we get:
y = arctan(x/7) - π/4
Therefore, the solution to the given initial-value problem is y = arctan(x/7) - π/4.